The Burgers equation (REF ) with \(\gamma =0\)  and \(\gamma =2\)  has received an extensive amount of attention since the studies by Burgers in the 1940s. If \(\gamma =0\) , the equation is perhaps the most basic example of a PDE evolution leading to shocks. If \(\gamma =2\) , it provides an accessible model for studying the interaction between nonlinear and dissipative phenomena. Kiselev et al. [1]} gave a complete study for general \(\gamma \in [0,2]\)  for the periodic case. In particular, for the case \(\gamma =1\) , they proved the global well-posedness of the equation in the critical Hilbert space \(H^{\frac{1}{2}}(\mathbb {T})\)  by using the method of modulus of continuity. Subsequently, Miao-Wu [2]} proved the global well-posedness of the critical Burgers equation in critical Besov spaces \(B^{1/p}_{p,1}(\mathbb {R})\)  with \(p\in [1,\infty )\)  with the help of Fourier localization technique and the method of modulus of continuity. For more results on the fractional Burgers equation and dispersive perturbations of Burgers equations, we refer the readers to see [3]}, [4]}, [5]}, [6]}, [7]} and the references therein. We should mention that Molinet et al. [7]} proved that the Cauchy problem for a class of dispersive perturbations of Burgers equations is locally well-posed in \(H^s(\mathbb {R})\) .
